A multiscale strategy for Bayesian inference using transport maps

TL;DR

This paper presents a multiscale Bayesian inference method that leverages transport maps and conditional independence to efficiently solve high-dimensional inverse problems, demonstrated on subsurface flow models.

Contribution

It introduces a novel multiscale decomposition using optimal transport maps to decouple coarse and fine-scale Bayesian inference, reducing computational cost.

Findings

Efficient inference on high-dimensional parameters (over 10,000)

Comparable accuracy to full-dimensional MCMC on moderate problems

Significant computational savings demonstrated in subsurface flow applications

Abstract

In many inverse problems, model parameters cannot be precisely determined from observational data. Bayesian inference provides a mechanism for capturing the resulting parameter uncertainty, but typically at a high computational cost. This work introduces a multiscale decomposition that exploits conditional independence across scales, when present in certain classes of inverse problems, to decouple Bayesian inference into two stages: (1) a computationally tractable coarse-scale inference problem; and (2) a mapping of the low-dimensional coarse-scale posterior distribution into the original high-dimensional parameter space. This decomposition relies on a characterization of the non-Gaussian joint distribution of coarse- and fine-scale quantities via optimal transport maps. We demonstrate our approach on a sequence of inverse problems arising in subsurface flow, using the multiscale finite element method to discretize the steady state pressure equation. We compare the multiscale strategy with full-dimensional Markov chain Monte Carlo on a problem of moderate dimension (100 parameters) and then use it to infer a conductivity field described by over 10,000 parameters.